Fro Last Tie Newton s three laws of otion: 1) Law of inertia ) F=a ( or a=f/ ) 3) Action and reaction (forces always coe in pairs Question If an apple falls toward the Earth, why doesn t the oon fall toward the Earth? A. The oon is too big B. The oon is too far away C. The oon does fall toward the earth. Velocity of the oon What is the direction of the Velocity of the oon? C A B Acceleration = change in velocity change in tie Velocity at tie t 1 Velocity at tie t Speed is sae, but direction has changed Velocity has changed How has the velocity changed? Velocity at tie t 1 Velocity at tie t V(t ) V(t 1 ) Change in velocity Centripetal acceleration = v /r, directed toward center of orbit. r = radius of orbit Earth s pull on the oon The oon continually accelerates toward the earth, But because of its orbital velocity, it continually isses the Earth. The orbital speed of the oon is constant, but the direction continually changes. Therefore the velocity changes with tie. True for any body in circular otion 1
Experient Newton s falling oon F= g F= g Acceleration of ball 1 =F/ 1 = g/ 1 1 accelerates inward in response to force g Acceleration = v /r for circular otion Throwing the ball fast enough results in orbital otion Fro Newton s Principia, 1615 Question A newton apple falls fro a tree. What is the net force on the apple? newtons downward What is the acceleration? g~10 /s downward Question, part I throw the N apple horizontally. After I throw it, what is the net force on the apple? N downward What is the acceleration of the apple? g~10 /s directly downward This is because, after I throw the apple, gravity is the only force. The only acceleration is due to gravity. Shoot the onkey After the dart leaves the gun, the only force is fro gravity. The only deviation fro straight-line otion is an accelaration directly downward. This is the sae as the onkey. Acceleration of oon So the oon is accelerating at directly toward the earth! This acceleration is due to the force of gravity. Is this equal to g, the acceleration of the apple? Can calculate it directly fro oon s orbital speed, and the Earth-oon distance. v r /s
The radius of the earth Distance and dia. of oon Originally fro study of shadows at different latitudes by Eratosthenes! R(earth)=6500 k The diaeter of the oon is the diaeter of its shadow during a solar eclipse. Fro the diaeter d and angular size d/r~5 deg, infer distance r~60*r(earth). Moon acceleration, cont Distance to oon = 60 earth radii ~ 3.84x10 8 Speed of oon? Circuference of circular orbit = πr orbital distance = πr Speed = =103 /s orbital tie = 7.3 days Centripetal acceleration = 0.007 /s This is the acceleration of the oon due to the gravitational force of the Earth. Distance dependence of Gravity The gravitational force depends on distance. Moon acceleration is 9.81 /s 0.007 /s 3600 ties saller than the acceleration of gravity on the Earth s surface. The oon is 60 ties farther away, and 3600=60 So then the gravitational force drops as the distance squared Newton: I thereby copared the force requisite to keep the Moon in her orb with the force of gravity at the surface of the Earth, and found the answer pretty nearly. Equation for force of gravity (Mass of object 1) (Mass of object ) F gravity square of distance between the F 1 d For asses in kilogras, and distance in eters, F = 6.7 10-11 1 d Exaple Find the acceleration of an apple at the surface of the earth Force on apple = F apple = 6.7 10-11 Earth apple d Acceleration of apple = F apple = 6.7 10 11 Earth apple d This is also the force on the Earth by the apple! d = distance between center of objects ~ radius of Earth = 6.67 10 11 N /kg 5.98 104 kg 6.37 10 6 = 9.83 /s ( ) 3
Gravitational force decreases with distance fro Earth Force on apple = F apple = 6.7 10-11 Earth apple d So oving farther fro the Earth should reduce the force of gravity Typical airplane cruises at ~5 i = 8000 d increases fro 6,370,000 to 6,378,000 Not noticeable! International space station orbits at 350 k = 350,000 d = 6,370,000 + 350,000 = 6,70,000 Again d has changed only a little, so that g is decreased by only ~%. So why is everyone floating around? The space station is falling just like Newton s apple Edward M. (Mike) Fincke, Expedition 9 NASA ISS science officer and flight engineer, is pictured near fresh fruit floating freely in the Zvezda Service Module of the International Space Station. (NASA) Jaes S. Voss, Expedition Two flight engineer, looks over an atlas in the Zvezda Service Module. (NASA) In its circular orbit, once around the Earth every 90 inutes, it is continuously accelerating toward the Earth at ~9.8 /s. Everything inside it is also accelerating at that sae rate. The astronauts are freely falling inside a freely-falling elevator. A little longer ride Supree Screa - 300 feet of pure adrenaline rush A freefall ride d = 1 at t = d a = 300 ft 3 ft /s = 4.3 sec of freefall Parabolic path of freely falling object 4
Question Halfway to the oon, what is the acceleration of an apple due to the Earth s gravity? But gravitational force is only % weaker - Need to ove farther fro the Earth. A. g/ B. g/4 C. g/900 Moon is 60 Earth radii fro the Earth. Halfway is 30 Earth radii. So apple is 30 ties farther than when on surface. Gravitational force is (30) ties saller = g/900 Acceleration of gravity on oon On the oon, an apple feels gravitational force fro the oon. Earth is too far away. Force on apple on oon = F apple = 6.7 10-11 oon apple r oon Accel. of apple on oon = F apple = 6.7 10 11 oon apple r oon Copare to accel on Earth = 6.7 10 11 Earth r Earth accel. on oon accel. on Earth = / oon Earth ( r oon /r Earth ) Accel. of gravity on oon accel. on oon accel. on Earth = oon / Earth ( r oon /r Earth ) = 7.4 10 kg/6.0 10 4 kg ( 1.7 10 6 /6.4 10 6 ) = 0.013 (.65) = 0.175 1 6 Gravitational force between sall objects Gravitaty force at large distances: Stars orbiting our black hole At the center of our galaxy is a collection of stars found to be in otion about an invisible object. 5
Orbits obey Newton s gravity, orbiting around soe central ass What is the central ass? http://www. pe.pg.de/ww w_ir/gc/intro. htl One star swings by the hole at a iniu distance b of 17 light hours (10 A.U. or close to three ties the distance to Pluto) at speed v= 5000 k/s, period 15 years. Fro the orbit we can derive the ass. The ass is.6 illion solar asses. It is ostly likely a black hole! 6